As it happens, there are more Democrats than Republicans in the Senate. Since in the end both houses must agree on a budget, perhaps the actual thing to do is to give each party the same number of votes, divided in some way between their senators and representatives.
As for whether in fact every year the budget would change in one direction, that assumes that cuts will continue to need to be made. It's not clear how you determine the level of cuts (or increases) to be made under this game. Also, the notion that the party in power has more influence on the budget seems to me to be a benefit from the perspective of democracy, rather than a negative.
I agree that granularity is a potential problem.
The US Congress is trying to resolve the national debt by getting hundreds of people to agree on a solution. This is silly. They should agree on the rules of a game to play that will result in a solution, and then play the game.
Here is an example game. Suppose there are N representatives, all with an equal vote. They need to reduce the budget by $D.
What game-theoretic problems does this game have? Can you think of a better game? Is it politically better to call it a "decision process" than a game?
The main trouble area, to my mind, is order of play. First I said that budget items would be listed by taking turns. The 1..N, N..1 order is supposed to make neither first nor last position preferable. But taking turns introduces complications, of not wanting to reveal your intentions early.
Then I said votes are placed secretly and revealed all at once. This solves problems about game-theoretically trying to conceal information or bluff your opponent. It introduces other problems, such as tragedy-of-the-commons scenarios, where every Republican spends their "defend" votes on some pork in their state instead of on preventing tax cuts, because they assume some other Republican will do that.
Is it better to play "cut" votes first, reveal them, and then play "defend" votes?
Is there a meta-game to use to build such games?